REVIEW
Negative thermal expansion:a review
W.Miller ÆC.W.Smith ÆD.S.Mackenzie ÆK.E.Evans
Received:4March 2009/Accepted:15June 2009/Published online:2July 2009ÓSpringer Science+Business Media,LLC 2009
Abstract Most materials demonstrate an expansion upon heating,however a few are known to hibit a negative coefficient of thermal expansivity (NTE).This naturally occurring phenomenon has been shown to occur in a range of solids including complex metal oxides,polymers and zeolites,and opens the door to composites with a coefficient of thermal expansion (CTE)of zero.The state of the art in NTE solids is reviewed,and under-standing of the driving mechanisms of the effect is con-sidered along with experimental and theoretical evidence.The various categories of solids with NTE are explored,and experimental methods for their experimental charac-terisation and applications for such solids are propod.An abstraction for an underlying mechanism for NTE at the supramolecular level and its applicability at the molecular level is discusd.
Introduction
In general,solids expand upon hey exhibit positive coefficients of thermal expansivity (CTE),denoted as a herein.However,a minority of solids show the inver f contracting upon heating,and thus exhibit negative thermal expansion (NTE).There has been an increasing amount of interest in the solids and their potential applications.The underlying mechanisms for NTE have been found to be complex.气魄的意思
The reason that most solids have positive CTEs is well understood which is due primarily to an increa in the interatomic bond length,which manifests at the macro-scopic level as an overall increa in a dimension or vol-ume.Bond lengthening is perhaps best explained by the potential energy versus interatomic distance diagram,e Fig.1.On heating,the vibrational energy ris,and due to the asymmetry of the potential energy curve,shown in Fig.1(which can be considered typical of most strong bonds),the mean interatomic distance increas [1].The rate of change (slope)of the potential energy curve is lower on the lengthening side of the curve than on the shortening side;thus,the mean bond length tends to increa with temperature.The so called ‘stronger’bonds have steeper and narrower potential wells resulting in a slower rate of increa in interatomic distance,and hence a smaller thermal expansion coefficient (a ).The CTE a is a measure of volumetric (a v )or linear (a l )change with temperature and is defined as a V ¼D V V 0D T
ð1Þa l ¼
D l l 0D T
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ð2Þ
where D V ,D l are the changes in volume and length,respectively,V 0,l 0are the initial volume and length,respectively,and D T is the change in temperature.In crystalline solids,a V can be split to show the extent of expansion/contraction of individual crystal axial directions.In the ca of isotropic solids,Eqs.1and 2are related by a v =3a l .However,in anisotropic solids,the relationship between a l and a V is not so simple as each crystal axis potentially has a different magnitude and sign of a giving three distinct values,a a ,a b and a c ,contributing to a V .
外相W.Miller ÁC.W.Smith (&)ÁD.S.Mackenzie ÁK.E.Evans School of Engineering,Computing and Mathematics,University of Exeter,Exeter EX44QF,UK e-mail:c.w.smith@ex.ac.uk
J Mater Sci (2009)44:5441–5451DOI 10.1007/s10853-009-3692-4
Several solids have now been identified with NTE behaviour,some associated with pha transitions and some stable over large temperature ranges.Examples include metal oxides [2–4],zeolites [5],Alp
Os [6,7],metal-organic frameworks [8]and other solids including well-known polymers [9–11]and fibres [12,13].The molecular and supramolecular structures of the solids are primary in determining whether the CTE is positive or negative [14].
Origin of negative thermal expansion
The NTE aris generally from supramolecular structural mechanisms,which dominate the erstwhile positive CTE of the interatomic bonds.No examples of solids with NTE in which the interatomic bond lengths shorten upon heating are known.In all the cas so far,the underlying mecha-nism for NTE is ascribed to higher supramo-lecular effects such as ferroelectric,magnetostrictive and displacive pha transitions,or low frequency phonon modes [15],the prence of rigid unit modes (RUMs)and libration.Ferroelectric pha transitions involve the ordering of dipole moments within a crystal structure.This
occurs in BaT iO 3when in the ferroelectric pha,due to slight distortions in the TiO 6octahedra,the dipole moments are aligned (ordered)[16].In the paraelectric pha (random),the dipoles are randomly orientated.Similarly,in transitions between ferro-and paramagnetic phas of a structure,the electron spins change,respec-tively,from being aligned to randomly orientated.Displa-ci
ve pha transitions [16]occur frequently in metal oxide solids such as ZrV 2O 7and quartz [17],whereby each atom in the crystal cell moves by a small amount relative to the surrounding atoms.The transitions do not involve any bond breaking or forming but can nevertheless significantly alter the structure between higher and lower symmetry forms.This is often manifested by rotations of rigid poly-hedra units.However,the majority of structures found to exhibit NTE have a common feature which is the two-coordinate (planar)M–O–M or Si–O–Si linkage.The linkages are typical in metal oxides such as ZrW 2O 8-type solids and in zeolites,and are key to their NTE behaviour.Bonds and associated vibrations
A phonon is a quantized mode of vibration,and may have a range of different frequencies,wavelengths and ampli-tudes,and be in or out of pha with each other.Two important phonon modes are the longitudinal and trans-ver vibrations depicted in Fig.2.Upon heating the lon-gitudinal vibration,modes tend to increa bond length,as explained above.In particular,M–O bonds increa in length which may,in turn,themlves promote an increa in M ÁÁÁM interatomic distances.In contrast,the transver vibration modes can have an opposing effect on the M ÁÁÁM distance.This is sometimes given a ‘guitar string’analogy,whereby,despite an increa in M–O bond length,the effective M ÁÁÁM distance decreas due to the change in M–O–M angle caud by the i
6669ncrea in amplitude of the oxygen atom’s vibration.At this point,it is worth noting that transver vibrational modes have lower excitation energy than longitudinal modes,and hence,they are exci-ted at lower temperatures,which,in turn,means they more often dominate at lower temperatures.The respon of
lub
the
Fig.2Schematic showing on the left ,longitudinal vibrations of M–O–M links found in network structures,and on the right ,transver vibrations responsible for NTE in some structures
M ÁÁÁM length is then a question of whether,in this ca,the longitudinal or transversal modes domi
nate.It appears,in many open framework structures,that the modes are the origin of NTE at lower temperatures.The extent of this lower temperature range is determined by other factors.长沙会计培训
The phonons modes are related to a via the Gru
¨nein parameter c [18]:a V ¼
c C V K V
ð3Þ
where C v ,K and V are the specific heat capacity,isothermal compressibility (also known as the bulk modulus),and
volume,respectively.The Gru
¨nein parameter,c ,usually ascribed to a crystalline solid,reflects the anharmonicity of the various phonon modes prent at a given temperature;negative values of which indicate that heating will produce a volumetric NTE,since all other parameters are constrained to be [0.The relationship of c to particular phonon modes is c ¼
À1dx 2ð4Þ
where x is the frequency of the mode and s is an applied strain (zero in the ca of a free standing solid).Phonon modes which show a decrea in frequency x as volume decreas have a negative c .Thus,a negative c substituted into Eq.3gives an overall negative contribution to a ,and hence a negative a [19].Normal anharmonicity in simple atomic pair phonons as shown in Fig.1is reflected by a positive value for c .More complex multi-atom phonons as shown in Fig.2may have negative values for c .
Libration
There is a characteristic of variable temperature XRD that can show a perceived contraction in M–O bond lengths,when plotted against temperature.Classical asymmetric potential wells indicate that all the bonds expand with an increa in temperature,albeit at different rates due to the varying strengths of different bonds.A weaker bond will expand faster than a stronger bond.However,in diffraction experiments,a contraction in the Si–O bond length is often obrved in silicates.This effect has been attributed to libration [19,20].Libration is a particular vibrational motion in a specific direction relative to the M–O bond.The XRD picks up an apparent shorter distance between M and O
and,therefore,does not give the true bond length.The librational effect is shown schematically in Fig.3[21].If we consider the tup where two atoms A and B are bonded at a fixed length,R ,lying along the x -axis,which denotes the mean orientation due to the mean positions of atoms hAi and hBi,then the bond librates about the O x
leha
direction such that atoms A and B move as indicated in Fig.3.
Furthermore,as the libration increas with temperature,the apparent bond length will decrea.It is worth noting at this stage that the apparent bond length will always be smaller than the true bond length.Furthermore,libration should not be confud with transver vibrations,which although appear to be similar,are in fact different types of vibrations.However,it appears from current experimental data for zeolites that strong libration effects often accom-pany Si–Si contraction and transver vibrations.In some cas,strong libration can show very negative apparent coefficients of thermal expansion a Si–O apparent,bad on the change in apparent Si–O bond length with temperature.Rigid unit modes
Structures consisting of corner-linked rigid polyhedra such as MO 4are able to rotate about the so-called ‘hinges’,of the type M–O–M or M–O–M 0linkages,where M and M 0are metal cations.The poly
gays
hedra are much have much higher frequency phonon modes,and so are relatively rigid in comparison with the M–O–M hinges which exhibit much lower frequency phonon modes.The modes are known as rigid unit modes (RUMs).In energy terms,the bending or rotation about the M–O–M linkage is many times more favourable than distortion of the polyhedral [22].This is due to interatomic repulsions between electro-negative oxygen atoms being very strong.The M–O–M hinges rotate due to the anharmonicity in their lateral vibrations as described by the Grunein parameter above.The so-called quasi RUMS (QRUMs)are very similar to RUMs,but have merely low frequencies instead of near-zero frequencies.The effect of this is that the QRUMs have higher energy and unlike RUMs exhibit polyhedral rota-tions and slight distortions in some or all the rigid polyhe-dra.This was demonstrated in ZrV 2O 7by Pryde et al.[23],where rotations of two-linked tetrahedral VO 4caud small but significant distortions within the octahedral ZrO 6units.Rigidity of polyhedra is mainly attributed to anion–anion
(oxygen–oxygen)repulsions.On increasing the cation size, the tetrahedra or octahedra increa in size,thus oxygen–oxygen distances increa resulting in a decrea in repul-sion[24].The reduced interaction strength in oxygen repulsions decreas the rigidity of the polyhedra allowing for distortions.The effect of this on NTE is shown in the A2M3O12-type structure(M=W and A=most cati
ons which can accept octahedral coordination),where increas-ing size of cation‘A’incread the effect of NTE[24,25]. For rotations of the polyhedra in the A2M3O12-type struc-ture to occur,there must be slight distortions in the poly-hedra.Thus,increasing cation size increas volume contraction.Rotations of polyhedra combined with poly-hedral distortions are also en in NASICON structures (NaZr2(PO4)3).The QRUMs in the NASICON structure manifest themlves as rotations of rigid tetrahedra causing the octahedra to distort as the temperature is incread[26].
equipmentIn a computational study by Hammonds et al.[27,28], the phenomenon of local RUMs was discusd.Local RUMs were propod as an integral part of structural flexibility,and are interesting in that they can result in localid thermally driven contraction of specific ctions or cages within a framework structure.Local RUMs are esntially the same as RUMs,in that the rigid units rotate about M–O–M linkages thus causing a structural change. The main difference is that unlike RUMs,local RUMs are not prent uniformly throughout the whole structure but are periodic.This phenomenon may have some special us since local thermally driven contraction may alter local physico-chemical properties of the atalytic activity.The local RUMs have been ud to explain catalytic activity of certain zeolites,where non-framework molecules induce a change in the local structure[29]. Pha transitions
The RUMs cau changes in the M–O–M angles as explained above;however,some structures,despite posssing rigid polyhedra,do not show this cooperative rotation of linked polyhedra giving ri to NTE behaviour.This is due to the way in which they are arranged in the larger structural net-work.This can be en in many structures which undergo a pha transition,at a particular temperature—a good example being the structure type A2(MoO4)3and the zeolite ferrierite[30].It was shown[31]on a study of Sc2(MoO4)3 that the monoclinic pha had a positive a V,and the ortho-rhombic pha had a negative a V.It was also shown[32]that generally such molybdate structures show positive a V in the monoclinic pha;however,after a pha transition to an orthorhombic unit cell,a V changes to negative.Symmetry is important in NTE and has been shown in many structures with displacive pha transitions,where a change in sym-metry is accompanied by a change in the sign of a.
During pha transitions the RUMs act as‘soft modes’which have a low and strongly temperature-dependent frequency.Soft modes may also be described as‘particular excitation which becomes unstable as the system approa-ches,the stability limit’[33].Whilst there is no specific limit to the number of‘soft modes’,usually there is one particular mode responsible for the pha change.As the temperature approaches,the pha transition temperature, Tc,the‘soft mode’frequency approaches zthelittleprince
ero,and at Tc, the system is unstable and undergoes a pha change to a stable form.The displacive pha from one ordered crystal structure to another,can be split into two types,namelyfirst-order discontinuous(a/b-quartz transi-tion)and cond-order continuous(ferrierite[30]which changes symmetry from Pmmn to Immm)transitions. The transitions involve some change in change in space group or crystal system[34],and may also involve a large reduction or increa in cell volume. However,in mostfirst-order transitions there occurs a rapid increa in volume,which may subquently be followed by contraction.A good example of this is the framework structure ZrV2O7.It was shown[35]that over the tem-perature ranging from-263to470°C,there was a pha transition which corresponded to a change in a sign.It was suggested[23]that the existence of quasi-RUMs(QRUMs) was the reason for NTE in the high temperature pha. Pha transitions reprent an interesting ca of NTE,but given their very specific temperature dependencies,they do not have the wide range of applications of the more stable, wider temperature range NTE similar to other solids. Microcracking
Anisotropic contraction can increa the apparent magni-tude of reduction in volume of bulk samples when per-forming dilatometry measurements.This was reported on the metal oxide Sc2(WO4)3[36],where the ceramic bar showed an approximatelyfivefold increa in contraction com
pared to the powder ND-determined thermal expan-sion.The reason why this occurs is not fully understood; however,it is clear that some macroscopic mechanism combines with a molecular mechanism to enhance the NTE effect.This has been postulated as being primarily due to microcracking of a compacted bulk sample.However, recent study by the authors has suggested that mere structural disorder,as would be caud by microcracking, cannot by itlf give ri to an NTE effect at the macro-scopic size scale[37].It is more likely that the compaction process gave ri to very specific packing and/or orienta-tion of crystals and apparent amplification of an already anisotropic NTE in one axis with conquent diminution of NTE in the other axes.Notably,this effect is less likely to happen in cubic solids having isotropic NTE behaviour.
A generalid mechanism
A generalid mechanism for NTE can be demonstrated schematically using the simplified 2D diagram in Figs.4and 5,for the perovskite and zirconium tungstate solids,respectively.The generalid mechanism requires RUMs to be prent,with suitably connected polyhedra and voids into which they are free to rotate.This depends upon the cooperative effects of the factors described in the ctions above.In Fig.4,the shaded squares show a slice through linked octahedra of MO 6as en in perovskite [38].Figure 5shows the xy -plane with rotations of linked octa-hedra about the z -a
xis.As can be clearly en in both of the structures due to rotations of the octahedra,at the M–O–M angle,the unit cell (outlined in the figures)decreas in area,and voids between units alter significantly from squares to parallelograms.The angle h (tilt angle)repre-nts the degree of rotation/change in M–O–M angle.In 3D,this is repeated in the other planes resulting in a decrea in volume.
This cooperative rotation of rigid corner linked units also occurs in structures of linked tetrahedra [39]or a mixture of linked tetrahedra and octahedra [27,31].Rotation of the M–O–M angle is finite,in that there is a limit to the change in angle,due to intermolecular forces and interactions.This limit varies depending of type of polyhedra and the local constraint in effect for example different connectivity within the network structure.Conquently,issues such as framework connectivity can restrict and even prevent rotations of the linked polyhedra taking place.Lightfoot et al.[6]studied veral zeolite structures,and found by comparison that many microporous zeolites show NTE,but despite similar structural units others do not,for example the PTE zeolite CIT-5and AlPO 4-31.A framework struc-ture or chiral symmetry posssing rigid units ems to be necessary but not sufficient for NTE behaviour to occur,as other as yet undefined structural conditions are equally necessary.The geometrical origin of NTE shown by Heine et al.[40],demonstrated some mathematic
al aspects of rotations of rigid units in framework structures.One par-ticular finding not echoed by many other authors,was that polyhedral rotations occur throughout the phonon spectrum,and are not confined to low frequency modes.However,it was also pointed out that due to the harmonicity of the phonon modes as reprented by c is strongly dependent on x (the phonon frequency,in fact x 2),the lower phonon modes have a disproportionate effect.This was shown in the study on b -quartz [17].
Returning to the statement at the beginning of this ction,when looking at NTE,there are esntially two mechanisms active during NTE.The thermal expansion between atom pairs has a positive contribution,whereas the geometric rotations of rigid units (transver vibrational modes and libration)has a negative contribution.For NTE
Fig.4Schematic of the
perovskite structure shown as octahedral units and 2D
reprentation shown as rotating squares,indicating the rotation mechanism
Fig.5Schematic of Zirconium Tungstate structure shown by tetrahedral and octahedral units in 3D [58]and the RUM mechanism in 2D
